# Representation of multi-variable functions as a composition of 1- or 2-variable functions [duplicate]

This is a re-post of my question from M.SE that remains unanswered for several months.

I'm familiar with Kolmogorov–Arnold representation theorem, but AFAIK their construction makes essential use of non-smooth functions. I am interested in what happens if we add the smoothness requirement.

There are some well-known special functions of 3 or more complex variables that are analytic in some domain (a region in $\mathbb C^n$) with respect to each variable. To give some examples: the incomplete beta function $B(z; a, b)$, the Lerch transcendent $\Phi(z, s, a)$, the Weierstrass elliptic function $\wp(z;g_2,g_3)$, hypergeometric-family functions, etc. Let's restrict their domain to reals.

It is possible to express all or some of these functions as a composition of several infinitely differentiable (with respect to each variable) functions of 1 or 2 real variables?

A similar question applies to functions of 2 variables (e.g. polylogarithms, incomplete elliptic integrals, Hurwitz zeta function, Bessel-family functions, etc.): Is it possible to represent them as a composition of several infinitely differentiable functions of 1 variable and the single fixed function of 2 variables $(x,y)\mapsto x+y$?

To give an example when the answer to the last question is positive, consider the complete beta function $B(a,b)$. It can be represented as $$B(a,b)=\exp\big[(\ln\Gamma(a)+\ln\Gamma(b))+(-\ln\Gamma(a+b))\big]$$ that is a composition of the 2-variable sum function and several infinitely differentiable 1-variable functions $x\mapsto\exp(x)$, $x\mapsto\ln\Gamma(x)$ and $x\mapsto -x$.

• Did you see the answer to mathoverflow.net/questions/140859/… and read the corresponding papers of Vitushkin (from 1954 and 1964)? In principle, it is known that for smooth functions the problem gets much harder. Feb 21 '16 at 19:20
• @VladimirDotsenko Thanks, I looked exactly for that answer. Feb 22 '16 at 1:16